Technical stuff: under the standard assumption of infinite exchangeability of coin tosses, there exists some limiting relative frequency for coin toss results. (This is de Finetti's theorem.)

Key point: I have a probability distribution for this relative frequency (call it f) -- not a probability of a probability.

You only have a small amount of information on the coin, and you decide for whatever reason that there's a 51% chance of getting heads. So you're going to bet on heads. But then you realize that there's a way to get more data.

Here you've said that my probability density for f is dispersed, but slightly asymmetric. I too can say, "Well, I have an awful lot of probability mass on values of f less than 0.5. I should collect more information to tighten this up."

"Gee, 51% isn't very high. I'd like to be at least 80% sure. Since I don't know very much yet, it wouldn't take much more to get to 80%. I should get that data so I can bet on heads with confidence."

This mixes up f on the one hand with my distribution for f on the other. I can certainly collect data until I'm 80% sure that f is bigger than 0.5 (provided that f really is bigger than 0.5). This is distinct from being 80% sure of getting heads on the next toss.